on Your Own An outfielder throws a is modeled ford home plate. The height of the ball above the ground (in feet) until it hits the he band is modeled by the function h(t)=-16t^2+30t+5 where I represents the time is seconds since the ball was thrown. Define and interpret the domain and range of the function. courresponding range and describe the end behavior

To define and interpret the domain and range of the function $h(t)=-16t^{2}+30t+5$, we need to consider the context of the problem.<br /><br />The function represents the height of the baseball above the ground as a function of time. Since time cannot be negative, the domain of the function is all non-negative real numbers, or $t \geq 0$.<br /><br />To find the range, we need to determine the maximum height the baseball reaches before it hits the ground. This occurs at the vertex of the parabola represented by the function. The vertex occurs at $t = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic term and linear term, respectively. In this case, $a = -16$ and $b = 30$, so the vertex occurs at $t = \frac{30}{2(-16)} = \frac{15}{8}$ seconds.<br /><br />Plugging this value back into the function, we can find the maximum height: $h\left(\frac{15}{8}\right) = -16\left(\frac{15}{8}\right)^2 + 30\left(\frac{15}{8}\right) + 5$. Simplifying this expression, we get $h\left(\frac{15}{8}\right) = \frac{465}{64}$ feet.<br /><br />Therefore, the range of the function is all values less than or equal to $\frac{465}{64}$ feet.<br /><br />The end behavior of the function can be described as follows: as $t$ approaches positive infinity, the height of the baseball approaches negative infinity, indicating that the baseball will eventually hit the ground. As $t$ approaches 0, the height of the baseball approaches 5 feet, representing the initial height of the baseball when it was thrown.